Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity Original Research Article

In this paper we present constitutive models for nearly incompressible, transversely isotropic materials in finite hyperelasticity, particularly for reinforced rubber-like materials, which are of essential engineering interest. The theory is developed using a convected curvilinear coordinate system based on a mixed two-field displacement–pressure energy functional. Furthermore, an a posteriori error estimator without multiplicative constants is derived for non-linear anisotropic problems, which measures the discretization error in the first Piola–Kirchhoff stresses in the L2-norm by solving local Neumann problems with equilibrated tractions. Illustrative numerical examples demonstrate the anisotropic material behaviour of reinforced materials and the efficiency of using adaptive finite element methods.